Methods and apparatus for magnetic resonance imaging in inhomogeneous fields

ABSTRACT

Methods for providing practical magnetic resonance imaging systems that utilize non-homogeneous background fields, B 0 , as well as, possibly non-linear, gradient fields G 1 , G 2  to make non-invasive measurements to determine, among other things, a spin density function. Two types of non-homogeneous background fields are considered: background fields B 0  in which the function |B 0 | does not have a critical point within the field of view, and background fields B 0  such that the function |B 0 | has a single critical point within the field of view. In the first case, an MR-imaging device may be constructed by using the permanent gradient in the background field, B 0 , as a slice select gradient, so long as particular criteria are met. In the second case, magnets may be constructed so that |B 0 | has an isolated non-zero local minimum. Using selective excitation, one can excite only the spins lying in a small neighborhood of this local minimum.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims benefit to U.S. Provisional Application Nos.60/663,937 filed Mar. 21, 2005 and 60/737,283 filed Nov. 16, 2005.

GOVERNMENT SUPPORT

The present invention was supported by the National Science Foundationunder Grant Nos. NSF DMS99-70487, DMS02-07123, and DMS02-3705. Thegovernment may have certain rights in the invention.

FIELD OF THE INVENTION

The present invention relates to methods and apparatus for obtainingmagnetic resonance images in inhomogeneous magnetic fields and, moreparticularly, to imaging with a background field B₀ that varies both inmagnitude and direction, both with and without critical points.

BACKGROUND OF THE INVENTION

In the standard approach to magnetic resonance imaging one uses a strongbackground field that is as homogeneous as possible. Commercial MRimaging magnets are homogeneous, within the field of view, to about 1ppm. In “open” MRI systems, the field homogeneity is somewhat less, butstill in this general range. One can imagine a variety of situationswhere it might be useful to do magnetic resonance imaging with theobject placed entirely outside the magnet's bore. As a consequence ofRunge's Theorem, it is possible to design coils so that this externalfield is as homogeneous as one would like, in a given region of space.However, this requires a large expenditure of power and complicated,difficult to design arrangements of coils. On the other hand, withsimpler arrangements of permanent magnets or electromagnets, one canproduce a field, B₀, such that, in a given region of space, exterior tothe magnets, or coils: (1) The field is strong; (2) The direction of B₀varies in a small solid angle; (3) The level sets of |B₀| are smooth;and (4) The size of ∇|B₀| is not too large.

Several groups have considered problems of this sort. Generallyspeaking, the prior art uses pulsed gradients for spatial encoding, andrefocusing pulses to repeatedly refocus the accumulating phase in thedirection of the permanent gradient. These ideas are described in U.S.Pat. No. 4,656,425 to Bendel, as well as in U.S. Pat. No. 5,023,554 toCho and Wong. The idea is further developed by Crowley and Rose asdescribed in U.S. Pat. No. 5,304,930 and U.S. Pat. No. 5,493,225. Pulsedgradients are also used in SPRITE, though for different reasons, asnoted by Balcom et al., Single-point ramped imaging with T ₁ enhancement(SPRITE), J. Mag. Res. A, Vol. 123 (1996), pp. 131-134.

Another group considering such problems is that of Dr. Alexander Pinesat University of California at Berkeley. His work is described in therecent PNAS paper: “Three-dimensional phase-encoded chemical shift MRIin the presence of inhomogeneous fields” by Vasiliki Demas, DimitrisSakellariou, Carlos A. Meriles, Songi Han, Jeffrey Reimer, and AlexanderPines. Their approach is somewhat different in that they try to matchinhomogeneities in the B₁-field with that in the B₀-field in order toeffectively “cancel” them out. Their efforts are more directed towardsspectroscopy and they consider very small field gradients.

Still another group working on problems of this sort is that of BernhardBlümich at RWTH Aachen in Aachen, Germany. This group's work isdescribed in “The NMR-mouse: construction, excitation, and applications”by Blümich B, Blumler P, Eidmann G, Guthausen A, Haken R, Schmitz U,Saito K, and Zimmer G in Magn Reson Imaging. 1998, pgs 479-484. Theirapproach is again different from what is described herein in that ituses a stroboscopic acquisition technique. Though it is very good forspectroscopy of materials, it is too time consuming and SAR intensivefor in vivo applications.

The present invention addresses methods and apparatus for imaging insuch a field as described in a paper by one of the present inventors (CL Epstein) entitled “Magnetic Resonance Imaging in InhomogeneousFields,” Inverse Problems, Vol. 20, pages 753-780 (Mar. 19, 2004), thecontents of which are hereby incorporated by reference in theirentirety. Further refinements of this method are presented herein foracquiring data that lead to a fairly standard 2d-reconstruction problem.

SUMMARY OF THE INVENTION

Methods are described for providing practical magnetic resonance imagingsystems that utilize non-homogeneous background fields, B₀, as well aspossibly non-linear basic gradient fields G₁, G₂ to make non-invasivemeasurements to determine, among other things, a spin density function.Generally, two types of non-homogeneous background fields areconsidered:

-   -   a. Background fields B₀ in which the function |B₀| does not have        a critical point within the field of view.    -   b. Background fields B₀ such that the function |B₀| has a single        critical point within the field of view.

In the case of background fields B₀ without a critical point in thefield of view, an MR-imaging device may be constructed by using thepermanent gradient in the background field, B₀, as a slice selectgradient, so long as:

-   -   a. the direction (as opposed to magnitude) of the background        field does not vary too much within the field of view;    -   b. the strength of the background field remains large throughout        the field of view so that the local Larmor frequency may be        determined to a high degree of accuracy by the components of the        various fields parallel to B₀;    -   c. the level sets of the function |B₀| within the field of view        are expressible as graphs of smooth functions over a region        lying in a plane;    -   d. apparatus can be constructed to generate magnetic fields G₁,        G₂ so that the functions |B₀|, <B₀, G₁>, <B₀, G₂> define        coordinates within the field of view;    -   e. for parameters (η₁, η₂) lying in a certain range, apparatus        is available to generate the fields η₁G₁+η₂G₂.; and    -   f. apparatus is available for generating sufficiently        homogeneous RF-fields within the field of view.

The methods of the invention demonstrate that there are many practicalcircumstances where these criteria can all be met. An apparatus meetingsuch criteria can allow one to make non-invasive measurements that allowa reconstruction of the spin density function determined by a3-dimensional object. The measurement process using such an apparatus inaccordance with the invention includes the following steps:

placing the object to be imaged in an inhomogeneous background field B₀for a sufficient time for the nuclear spins of a desired species to bepolarized;

selectively exciting the polarized nuclear spins using standardselective excitation RF-pulse sequences from apparatus that generatessubstantially homogeneous RF-fields within a field of view of theapparatus; and

spatially encoding phases of the excited nuclear spins using gradientfields G₁, G₂, wherein the functions |B₀|, <B₀, G₁>, <B₀, G₂> definecoordinates within the field of view of the apparatus.

In exemplary implementations, a single refocusing pulse may be used todescribe both a pure frequency encoding scheme as well as a combinedphase encoding and frequency encoding scheme. In an exemplaryembodiment, the measurement process employs a 3-dimensional encodingscheme wherein a complete line in 3-dimensional k-space is read aftereach excitation and refocusing pulse.

In other exemplary implementations, the method includes the steps ofselecting a 2-dimensional slice of the object for excitation that is notperpendicular to G₀, exciting the 2-dimensional slice of the object byapplying a selective RF-pulse in the presence of a slice excitationgradient G_(ss), where G_(ss) is a linear combination of G₀ and atransverse gradient G_(ss) ^(app), generated as a linear combination ofbasic gradient fields G₁, G₂ provided by a magnetic resonance scanner,and applying a readout gradient G_(re) where G_(re) is a linearcombination of G₀ and a transverse gradient G_(re) ^(app) generated as alinear combination of G₁, G₂, where G_(ss) and G_(re) are not parallelat any point in the selected excited slice. The selected excited sliceof the object may then be reconstructed and displayed in a conventionalmanner. The selected excited slice may also be phase encoded with agradient G_(ph) generated as a linear combination of G₁, G₂. Thefunctions X=<G₁, B₀>, Y=<G₂, B₀> and Z=|B₀(x,y,z)|, may define localcoordinates that map the field of view of the magnetic resonance scanneronto a region of space, whereby G_(ss)=B₀+G₁ and G_(re)=B₀−G₁. The stepof applying the readout gradient G_(re) may also include the step ofapplying at least one refocusing pulse.

Since the spatially encoded signal may be interpreted as samples of theFourier transform of a function of three physical variables, an imagereconstruction function can be recovered using standard Fourierinversion methods. The methodology of the invention set forth hereinindicates that there is no difficulty, in principle, in obtaining asufficiently large signal to acquire useful measurements, even in thepresence of a permanent gradient in the B₀ field.

The selective excitation function may be shown to be the result ofapplying a coordinate change to the spin density function andmultiplying this function by a positive function. The change ofcoordinates and positive multiplier can both be determined by standardcomputations from the known information |B₀|, <G₁,B₀>, <G₂,B₀>. Thisdata is determined by the physical apparatus and, given that thehardware remains in calibration, need only be computed once and stored.With this information, and the image reconstruction function, thedesired spin density function can be reconstructed.

On the other hand, in the case of background fields B₀ with a singlecritical point in the field of view, the method of the inventiondescribes how magnets may be constructed so that |B₀| has an isolatednon-zero local minimum. Using selective excitation, one can excite onlythe spins lying in a small neighborhood of this local minimum. In thisway one can do spatially localized, high SNR, spectroscopicmeasurements, without any need for further spatial encoding.

BRIEF DESCRIPTION OF THE DRAWINGS

The systems and methods for generating magnetic resonance images ininhomogeneous fields in accordance with the present invention arefurther described with reference to the accompanying drawings, in which:

FIG. 1 illustrates an electromagnet constructed out of two concentriccylindrical segments where the inner cylinder subtends 270°, and theouter cylinder subtends 180°. As indicated, the FOV is a region of spaceexterior to the arcs of both cylinders.

FIG. 2 illustrates the properties of the magnetic field generated by thearrangement of conductors shown in FIG. 1, where the current on theinner cylinder is 0.6 units and the current on the outer cylinder is 2units. FIG. 2( a) illustrates level contours of |B₀,| and field vectors(the essentially vertical lines), while FIG. 2( b) illustrates a plot of|B₀,| (solid line) and ∇|B₀|/|B₀|, (dashed line) along y=0.

FIG. 3 illustrates a graph of a typical normalized magnetizationprofile.

FIG. 4 illustrates the cone that contains the samples points for F(ρ),showing some lines along which samples are acquired.

FIG. 5 illustrates the cylinder within the cone, in which the samplepoints for the regridded data lie.

FIGS. 6( a) and 6(b) respectively illustrate two level surfaces ofB_(0,εδ) with b₀=1, ε=0.1, δ=0.0025.

FIG. 7 illustrates slant slice imaging with an adjustable gradient ofequal strength to the permanent gradient.

FIG. 8 illustrates a timing diagram for the slant slice imaging methodshown in FIG. 7.

FIG. 9 illustrates an image created from measurements made with apermanent background field and the imaging sequence shown in FIG. 8.

FIG. 10 illustrates slant slice imaging with an adjustable gradient ofsmaller strength than the permanent gradient.

FIG. 11 illustrates level sets of G0+G1 (circular arcs) and G0−G1(hyperbolas), where a slice is a region between two circular arcs andthe slice averaging is along the hyperbolas.

FIG. 12 shows images made using the slant-slice protocol with variousvalues of the ratio

$v = {\frac{g_{x}}{g_{z}}.}$The geometric distortion due to the slant of the read-out gradient isnot corrected in these images.

FIG. 13 shows images from FIG. 12 with the corrections for the geometricdistortion.

FIG. 14 illustrates that when the ratio between G₀ and G₁ equals 1,their strengths are simultaneously increased.

FIG. 15 illustrates a “one-sided” 3d MR-imaging system designed usingthe techniques of the invention wherein the sample (patient) would lieon a patient table to one side of the magnet.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

Certain specific details are set forth in the following description withrespect to FIGS. 1-15 to provide a thorough understanding of variousembodiments of the invention. Certain well-known details are not setforth in the following disclosure, however, to avoid unnecessarilyobscuring the various embodiments of the invention. Those of ordinaryskill in the relevant art will understand that they can practice otherembodiments of the invention without one or more of the detailsdescribed below. Also, while various methods are described withreference to steps and sequences in the following disclosure, thedescription is intended to provide a clear implementation of embodimentsof the invention, and the steps and sequences of steps should not betaken as required to practice the invention.

Overview

The present invention relates to three problems that arise in magneticresonance (MR) imaging: (1) excitation, (2) spatial encoding, and (3)image reconstruction. There are, of course, many other issues that wouldarise in the practical implementation of such an imaging system.Nonetheless these issues must be addressed first.

First of all, one must understand what is meant by “imaging in aninhomogeneous background field” in accordance with the invention. At theoutset, the problem of imaging with an inhomogeneous background fieldsplits into two cases:

-   [Noncritical case] The function |B₀| has no critical points in the    field of view, the level sets of |B₀| are smooth, and fit together    nicely.-   [Critical case] The function |B₀| has critical points within the    field of view.    The techniques available for spatial localization are different in    each case and are treated separately.

Because the local resonance frequency is determined by the magnitude ofthe local field, small variations in |B₀| are of much greater importancethan small variations in its direction. In accordance with theinvention, imaging is performed in an inhomogeneous background field if∇|B₀| has a “large,” time independent component throughout the imagingexperiment. The inventors are not considering the sorts of “random” orlocalized inhomogeneities that arise from the physical properties of theobject being imaged, e.g. susceptibility artifacts. For much of thisapplication, the inventors assume that the function |B₀| i“noncritical,” i.e., has no critical points in the field of view, thelevel sets of |B₀| are smooth, and fit together nicely. One can write:B ₀ =B ₀₀ +G ₀,  (1)where B₀₀ is a constant uniform field, and the spatial variation of B₀around the constant background is captured by G₀. The choice of B₀₀ is,a priori, rather arbitrary. One usually selects B₀₀ as the value of B₀at a central point in the excited slice.

The first problems that one encounters are connected to selectiveexcitation. Because the permanent gradient tends to be large, a largeRF-bandwidth may be required to excite a sufficiently wide slice. Beyondthis bandwidth problem, one might also expect small variations in thedirection of B₀ to lead to difficulties with selective excitation.However, this turns out to be a relatively minor problem. It can beshown mathematically that if the direction of B₀ does not vary too much,then a selective RF-pulse sequence excites spins lying in a region ofspace of the form:{(x,y,z): ω₀ −Δω≦γ|B ₀(x,y,z)|≦ω₀+Δω}.  (2)If ∇|B₀| does not vanish within this set then the selected slice is anonlinear analogue of the region between two planes.

In the noncritical case, the permanent gradient in B₀ may be used as aslice select gradient. In this case, the problem of imaging in anoncritical, inhomogeneous field becomes, at least conceptually, theproblem of imaging with a permanent slice select gradient. This is notan imposed gradient, which could be reversed, but rather a permanent andirreversible gradient that arises from the basic design of the imagingsystem and the placement of the sample. In the noncritical case,conditions on the background field may be obtained so that the measuredsignal can, up to appropriate changes in variable, be regarded assamples of a Fourier transform. In principle, the needed changes ofvariable can be computed with a knowledge of the background field, andthe gradient fields throughout the field of view. The inventors'principal observation in this regard is that, so long as the directionof B₀ does not vary too much within the field of view, the mostimportant property for the imaging system is to have a 2-parameterlinear family of gradient fields. As explained below, the gradientsthemselves need not be linear.

As a concrete example, consider the following situation: one has twoinfinite, concentric arcs of cylinders with current running in oppositedirections, along their axes. The field of view is a region outside ofboth arcs. The setup is shown in FIG. 1. With infinitely long cylinders,the field is independent of the z-coordinate and lies in the xy-plane.FIG. 2( a) shows level lines of the field generated by the apparatus inFIG. 1, along with vectors indicating its local direction. FIG. 2( b)shows the field strength (solid line), and the relative gradient (dashedline) ∇|B₀|/|B₀|, along the y-axis.

At each point (x,y,z), the direction of the background field, B₀(x,y,z),defines a “local z-direction.” The transverse component of themagnetization at (x,y,z) is the part of the magnetization orthogonal toB₀(x,y,z), This component of the magnetization precesses aboutB₀(x,y,z), at the local Larmor frequency, which equals γ|B₀(x,y,z)|. Asthe definition of the “rotating reference frame” varies from point topoint, one may work in the laboratory reference frame.

For purposes of the present patent application, the perturbations of B₀,used for spatial encoding (i.e. gradients), are referred to as “timeindependent” fields. Such fields are typically turned on for a period oftime, and then turned off, and so are not, strictly speaking, timeindependent. With use this terminology to distinguish these fields fromthe RF-fields used for selective excitation. The RF-fields areconstantly varying on the time scale of the Larmor period. The reasonfor this distinction is that, so long as B₀ is a very strong field, theimportant component of a “time independent” field is the componentparallel to B₀, whereas the important components of an RF-field arethose orthogonal to B₀. Throughout this description, it is assumed thatthe B₀-field is strong enough that the components of gradient fields, indirections orthogonal to it, can safely be ignored.

B₀ is used to denote magnetic fields in the direction of B₀₀ In mostapplications of MR it is only the spatial dependence of the magnitude ofB₀ which is carefully accounted for. Indeed it is usually assumed thatthe gradients are “linear” so that (in standard homogeneous fieldimaging):B₀₀=(0,0,b₀)  (3)B ₀ =B ₀₀+<(x,y,z),(g _(x) ,g _(y) ,g _(z))>  (4)The linear function <(x,y,z),(g_(x),g_(y),g_(z))> (or sometimes thevector (g_(x),g_(y),g_(z)) itself) is called the “gradient.” It is theprojection of a gradient field in the direction of B₀. In theinhomogeneous case, one needs to be a bit more careful, and so theinventors work with the magnetic fields which generate the gradients.The present inventors distinguish between gradient fields, which arequasi-static magnetic fields used to induce perturbations in thebackground field and field gradients. Strictly speaking, the fieldgradient G generated by the gradient field G is defined to be:

$\begin{matrix}{G = {\nabla\left\langle {G,\frac{B_{0}}{B_{0}}} \right\rangle}} & (5)\end{matrix}$If |G₀|<<|B₀| and G₀ is not too rapidly varying, then the followingapproximate value may be used:

$\begin{matrix}{{G \approx {\nabla\left\langle {G,\frac{B_{00}}{B_{00}}} \right\rangle}},} & (6)\end{matrix}$for the field gradient, in the computations.

In the MR-literature, a point where ∇|B₀| vanishes is sometimes called a“sweet spot.” Following the usual practice in the mathematicsliterature, such a point is called a critical point of |B₀|. The valueof |B₀| at a critical point is called a critical value of |B₀|. Thecoordinates are normalized so that the critical point is located at(0,0,0) and if ω₀ denotes the critical value, then γ|B₀(0,0,0)|. Thegeometry of the level sets S_(ω), for values of ω near to ω₀, isdetermined by the nature of the critical point that |B₀| has at (0,0,0).This is explained in Section 5 of [9]. Because the components of B₀ areharmonic functions, the function |B₀(x,y,z)| is subharmonic. The maximumprinciple implies that |B₀| cannot have a local maximum value. Thus(0,0,0) can be either a saddle point or a local minimum. To theinventors' knowledge, the only case that has been considered in theliterature is that of a saddle point. This case is analyzed in detail inSection 6 of the aforementioned paper by Epstein, entitled, MagneticResonance Imaging in Inhomogeneous Fields, Inverse Problems, Vol. 20(2004), pp. 753-780, where a new explanation is given as to why this isa problematic geometry for imaging. In this document, examples areconstructed to show that magnetic fields exist such that |B₀| attains anonzero local minimum value. These fields provide new opportunities forlocalized spectroscopy, which are not available in earlier approaches ofthis sort, e.g. FONAR and TOPICAL. In particular, with each acquisition,one can measure a complete FID generated by the material located in asingle pixel.

The measurement process for background fields without critical pointswithin the field of view is considered below as well as how to constructfields so that |B₀| has an isolated minimum value.

Inhomogeneous Fields without Critical Points

Most approaches to magnetic resonance imaging in a homogeneousbackground field follow essentially the same sequence of steps:

-   1. The sample is polarized in the uniform background field.-   2. Using a slice select gradient, the sample is selectively excited    using an RF-pulse.-   3. By reversing the slice select gradient, or using a refocusing    pulse, the excited magnetization is rephased.-   4. Using gradient fields, the excited magnetization is “spatially    encoded.”-   5. The signal is acquired, possibly with additional spatial    encoding.

In this first embodiment, the problem of imaging with an inhomogeneousbackground field without critical points is considered. A simplifiedmodel, with a background field, B₀(x,y,z), of the form:B ₀(x,y,z)=(b ₀ −Gz){circumflex over (z)}  (7)is considered for z∈[−z_(max),z_(max)]. While G/b₀ may be large, perunit distance, the field of view is constrained so that b₀±Gz_(max) isalso assumed to be large. This means that B₀ is large within the fieldof view, and therefore, slowly varying perturbing fields, orthogonal toB₀, can safely be ignored. There are many possibilities for theplacement of coils to generate the gradients and RF-pulses needed to doimaging. In some circumstances, these coils could be placed around thesample, in other cases they could be placed on one side of the sample.For a preliminary analysis, it is assumed that the gradient in |B₀| isparallel to the direction of B₀. This simplifies the discussion, alittle, but is not necessary to do the analysis. A similar analysis alsoapplies to fields which are not assumed to point in a fixed direction,provided |B₀| is large throughout the field of view, and ∇|B₀| does notvanish. This more general case is considered below.

The following convention is used in this section: The Fourier transformof a function ƒ is denoted by F(ƒ). For the laboratory frame,{{circumflex over (x)},ŷ,{circumflex over (z)}}, with {circumflex over(z)} parallel to B₀., then:[a+ib,c]⇄a{circumflex over (x)}+bŷ+c{circumflex over (z)},  (8)with the complex number, a+ib, representing the transverse component ofthe magnetization. The computations in this section are done in theresonance rotating reference frame defined by b₀{circumflex over (z)}.

In the analysis of fields without critical points, the permanentgradient in B₀. may be used as a slice select gradient. Leaving thesample stationary in the background field produces an equilibriummagnetization, M₀(x,y,z), given by:M ₀ =C(T)ρ′(x,y,z)B ₀(x,y,z)  (9)

$\begin{matrix}{{C(T)} = {\frac{1.0075}{T}{Am}^{- 1}\mspace{14mu}{for}\mspace{14mu}{protons}\mspace{14mu}{in}\mspace{14mu}{{water}.}}} & (10)\end{matrix}$Here ρ′(x,y,z), is the density of water protons at (x,y,z). As noted,the constant C(T) is inversely proportional to T, the absolutetemperature. To simplify notation, ρ denotes C(T)ρ′. A principal goal ofMRI is the determination of the function ρ(x,y,z). If |B₀(x,y,z)| variesconsiderably over the support of ρ(x,y,z), then it may be necessary toinclude |B₀(x,y,z)| in the definition of the equilibrium magnetization.Such variation will result in a slowly varying shading of the image thatcan be removed by post-processing. In this section, it is assumed thatthis is not the case, and we use b₀ to denote |B₀(x,y,z)| in the formulafor the equilibrium magnetization.

In this embodiment, the slice select gradient does not have to be“turned on.” While the Bloch equation analysis of selective excitationapplies, essentially verbatim, a few remarks are in order. If B₀ alwayspoints in the same direction, as in equation (7), then the usualanalysis of selective excitation, from the homogeneous case, applieswithout change. In general, the direction of B₀. may vary slowly overthe field of view. In this case, a selective RF-pulse designed for usewith a B₀.-field having a permanent gradient, but uniform direction, maystill be used as in equation (7). It may be shown mathematically thatthe main consequence of non-orthogonality between B₀. and B₁ is adecrease in the effective amplitude of B₁. If the angle betweenB₀(x,y,z) and B₁(x,y,z;t) is

${\frac{\pi}{2} + \phi},$then the effective RF-field at this point is

$\frac{\left( {1 + {\cos\;\phi}} \right)}{2}{{B_{1}\left( {x,y,{z;t}} \right)}.}$Attenuating the RF slightly diminishes the flip angle and introduces asmall phase error, but has very little effect on the selectivity of thepulse. If, over the extent of the sample, the angle between B₁ and B₀ isclose to

$\frac{\pi}{2},$then there will be some (removable) shading in the image. Hence, if B₁is designed to excite spins with offset frequencies in the band aboutthe Larmor frequency, [ƒ_(min),ƒ_(max)], then the actual excited sliceis given by the (possibly nonlinear) region of space{(x,y,z): ω₀+ƒ_(min) ≦γ|B ₀(x,y,z)|≦ω₀+ƒ_(max)}.The slices are bounded by level sets of |B₀(x,y,z)|.

The first significant difference between imaging in homogeneous fieldsand in inhomogeneous fields occurs at step 3. In the latter case, it isnot possible to reverse the slice select gradient to rephase themagnetization after the application of a selective pulse. The onlyrealistic options are to use a self-refocused pulse or a refocusingpulse. A self refocused pulse, is only refocused once, so this option isnot considered further.

A normalized excitation profile is shown in FIG. 3. Assuming B₀ is givenby equation (7), then, at the end of a selective RF-pulse, themagnetization is given by:M(x,y,z)=Cb ₀ρ(x,y,z)[e ^(iγGzτ) ¹ w(z),sgn(z)√{square root over (1−w²(z))}]  (11)with τ₁ the rephasing time for the selective RF-pulse. The functionsgn(z) takes the values ±1; it is included to allow for flip angleslarger than 90°. In general sgn(z)=1 for z outside a finite band. If onewere to immediately apply a refocusing pulse, then, at its conclusion,the magnetization would be given by:M(x,y,z)=Cb ₀ρ(x,y,z)[e ^(−iγGzτ) ¹ w(z),sgn(z)√{square root over (1−w²(z))}]  (12)Normalizing so that t=0 at the conclusion of the refocusing pulse, themagnetization, as a function of space and time is given, for t≧0, by:M′(x,y,z;t)=Cb ₀ρ(x,y,z)[e ^(iγGz(t−τ) ¹ ⁾ w(z),sgn(z)√{square root over(1−w ²(z))}]  (13)and therefore:M′(x,y,z;τ ₁)=Cb ₀ρ(x,y,z)[w(z),sgn(z)√{square root over (1−w²(z))}]  (14)As has been known since the work of Hahn, one can create spin echoes,even with a permanent field gradient. The analysis is only slightlydifferent if the direction of B₀ is slowly varying. This leads to smallvariations in the degree to which the magnetization is refocused which,in turn, produces a slowly varying shading in the reconstructed image.

To measure the total density of spins within the slice, one couldaverage the signal over a time interval [τ₁−τ_(acq),τ₁+τ_(acq)] leadingto a measured signal of the form:

$\begin{matrix}{S_{echo} = {C\;\omega_{0}^{2}{\int_{o\; 3}{{{\sin c}\left( {\gamma\;{Gz}\;\tau_{acq}} \right)}{b_{{rec}\; 1}\left( {x,y,z} \right)}{\rho\left( {x,y,z} \right)}{w(z)}\ {\mathbb{d}x}{\mathbb{d}y}{{\mathbb{d}z}.}}}}} & (15)\end{matrix}$Here, it is assumed that τ_(acq)≦τ₁. Supposing that G is large, and Δzis the width of the excited slice, the requirement that sinc(γGzτ_(acq))remain positive throughout the excited slice leads to a maximumreasonable value for τ_(acq):

$\begin{matrix}{{\tau_{acq} \leq {\frac{\pi}{\Delta\; f}\mspace{14mu}{where}\mspace{14mu}\Delta\; f}} = {\gamma\; G\;{{\Delta z}.}}} & (16)\end{matrix}$Since the length of the time interval over which the signal is averagedeffectively determines the bandwidth of the measured signal, it may beseen that, in inhomogeneous field imaging, the size of the permanentgradient effectively determines the minimum “receiver bandwidth.” Thisquestion is analyzed at length in Section 4 of the Epstein paperreferenced above.

In step 4, the encoding of spatial information, it is apparent fromequation (13) that the phase of the transverse magnetization is marchinginexorably forward. This suggests a stroboscopic approach to signalacquisition. By combining gradients with refocusing pulses, one couldsample the Fourier transform of ρ. How often this procedure can berepeated is largely determined by the size of T₂, and how well themagnetization can be repeatedly refocused. If the permanent gradient islarge, then diffusion effects might also lead to a rapid decay of signalstrength. This approach to the problem has been considered by severalgroups of investigators. For the most part, the previous work usespulsed gradients for spatial encoding, and refocusing pulses torepeatedly refocus the accumulating phase in the direction of thepermanent gradient. These ideas are described in U.S. Pat. No.4,656,425, as well as in U.S. Pat. No. 5,023,554. The idea is furtherdeveloped by Crowley and Rose as described in U.S. Pat. Nos. 5,493,225and 5,304,930. Pulsed gradients are also used in SPRITE, though fordifferent reasons.

Crowley and Rose make the important observation that a large permanentgradient leads to a rapid traversal of k-space, and a short refocusingtime. Accordingly, within the time constraints imposed by transverserelaxation, many samples can be collected. On the other hand, a largegradient means that, to excite a reasonably sized slice, requires alarge RF-bandwidth and thereby an increased SAR. Indeed, as a refocusingpulse is required for each point sampled in k-space, these imagingsequences have a much larger SAR than most sequences used withhomogeneous fields. In the afore-mentioned Epstein paper, the presentinventor examined how SNR and SAR requirements limit the ratio|∇|B₀∥/|B₀|.

Alternative approaches to the problem of spatial encoding and signalacquisition, which use a single refocusing pulse per line in k-space,will now be described. We first describe an embodiment using a3-dimensional imaging protocol. Some applications of this idea lead toirregularly spaced samples, and so one might use a technique likeregridding to obtain regularly spaced samples, before reconstructing animage. An analogue of a phase encoding-frequency encoding method and aradial, pure frequency encoding method are described. The latterapproach is described first.

After the selective excitation, the magnetization is allowed to freelyprecess for an additional τ₂ units of time so that, after a refocusingpulse, the magnetization is given by:M(x,y,z)=Cb ₀ρ(x,y,z)[e ^(−iγGzτ) ³ w(z),sgn(z)√{square root over (1−w²(z))}]  (17)where τ₃=τ₁+τ₂. At this point, a gradient of the form:B _(fe0)=<(g _(x) ,g _(y),0),(x,y,z)>{circumflex over (z)}  (18)is turned on. If t is normalized so that the end of the refocusing pulseoccurs at t=0, then the transverse magnetization is given, for t≧0, by:M _(xy)(x,y,z;t)=Cb ₀ρ(x,y,z)e ^(iγ[Gz(t−τ) ³ ^()+t(g) ^(x) ^(x+g) ^(y)^(y)]) w(z)  (19)Sampling at times t∈{jΔt:j=0, . . . ,N}, one measures approximate valuesfor {F( ρ)(jΔk_(x),jΔk_(y),jΔk_(z)−k_(zmax))}, where:ρ(x,y,z)=Cb ₀ρ(x,y,z)b _(1rec)(x,y,z)w(z)  (20)andΔk _(x) =γg _(x) Δt,Δk _(y) =γg _(y) Δt,Δk _(z) =γGΔt.  (21)By adjusting the coefficients of the gradient field, B_(fe0), one canobtain samples of F(ρ) along straight lines lying within a cone, C, withvertex at (0,0,−γGτ₃), as shown in FIG. 4. By regridding, one can thenobtain samples of F(ρ) which are uniformly spaced on a cylindrical grid,lying inside a cylinder, K contained within the cone C, as shown in FIG.5. An image could then be reconstructed by using the standard Fouriertransform in the z-direction and a filtered back-projection algorithm inthe transverse plane.

To use a method like regridding requires a certain amount ofoversampling. For each sample point p_(j) on the regular grid within K,severally irregularly spaced samples must be collected in a neighborhoodof p_(j). It is also clear that samples from near the vertex of C maynot be usable in the regridding process. Additionally, the samplesbecome rather spread out as one crosses the k_(z)=0 plane. It may bepreferable, to measure F(ρ)(k), for k with nonpositive k_(z), andrecover the values in the other half plane using the conjugate symmetry.

It is clear that there are many possible variations on this generalapproach to spatial encoding. For example, one might begin with a largerτ₃ and a large initial gradient to first move the vertex of the cone to(k_(x0),k_(y0),k_(z0)). This constitutes a phase encoding step. Afterreducing the gradient, samples could then be collected along lines ofthe form: (k_(x0),k_(y0),k_(z0))+tγ(g_(x),g_(y),G).

Indeed, one could turn off the x and y gradients and collect samples ofthe Fourier transform of ρ along vertical lines in k-space. If theinitial points (k_(x0),k_(y0),k_(z0)) are on a uniformly spaced grid inthe (k_(x),k_(y))-plane, then a standard FFT could be used toreconstruct the image.

2-D Imaging Protocol

A second embodiment using a 2-dimensional imaging protocol will now bedescribed. In this case, the permanent gradient is employed, along witha field of the form G_(ss)=η₁ ^(s)G₁+η₂ ^(s)G₂, to define the sliceselection direction, and a field transverse to this direction, G_(re) asthe read out gradient. A third field of the form G_(ph)=λ(η₁ ^(p)G₁+η₂^(p)G₂) is used as a phase encoding gradient. To describe the invention,a simple case is first considered wherein all of the gradients arelinearly varying magnetic fields. As usual, it is assumed that thebackground field B₀ is sufficiently strong throughout D that componentsof fields orthogonal to B₀ have a very small effect on the measurementsand can safely be ignored.

Analysis in the Linear Case

Let us suppose that B₀₀=(0,0,b₀) and let:B ₀=(0,0,b ₀)+(*,*,g _(z) z)=B ₀₀ +G ₀G ₁=(*,*,x) and G ₂=(*,*,y)  (22)Here * is used to denote negligibly small field components orthogonal to(0,0,b₀). The permanent field gradient is of the form:G_(z)=(0,0,g_(z)),  (23)and we assume that one can generate fields η₁G₁+η₂G₂, with gradients ofthe form:G _(η)=η₁(1,0,0)+η₂(0,1,0),  (24)where |η_(j)|≦m_(g).Strong Transverse Gradients (G₀≦G₁)

The simplest case arises when m_(g)εg_(z). In this case, one uses forthe slice select gradient the field G_(ss)=g_(z)G₁+G₀, and for the readgradient, the field G_(re)=−g_(z)G₁+G₀. The phase encoding gradientfield is then G_(ph)=λ(*,*,y), where λ assumes values in the range[−m_(g),m_(g)]. The slice select field gradient is thenG_(ss)=(g_(z),0,g_(z)), while the read-out field gradient isG_(re)=(−g_(z),0, g_(z)). With these fields, the approach to imaging inaccordance with the invention is the following:

-   -   1. Place the sample in the static field, B₀, long enough to        polarize the nuclear spins.    -   2. Turn on the gradient field g_(z)G₁ to attain a slice select        field gradient G_(ss)=(g_(z),0,g_(z)).    -   3. Apply a selective RF-pulse to flip spins lying in the region        of space where:        ƒ₀ −Δƒ≦γ|B ₀ +g _(z) G ₁|≦ƒ₀+Δƒ,        leaving the spins outside this region essentially in their        equilibrium state. If ƒ₀=γb₀, then the region of space excited        is the slanted slice given by:        {(x,y,z): −Δƒ≦γg _(z)(x+z)≦Δƒ}.  (25)        If w(s) denotes a function that is 1 for −Δƒ≦s≦Δƒ and zero        outside a slightly larger interval, and τ₀ denotes the rephasing        time for the selective RF-pulse, at the conclusion of the        RF-pulse the transverse magnetization has the form:        m(x,y,z;0)=sin αρ(x,y,z)w(γg _(z)(x+z))e ^(iτ) ⁰ ^(γg) ²        ^((x+z))  (26)        Here α is the flip angle and ρ is a spin density function,        normalized to take account of the magnitude of the equilibrium        magnetization, and τ₀ is the rephasing time for the RF-pulse.    -   4. At the conclusion of the RF-pulse, the adjustable gradient        g_(z)G₁ is switched off and the excited magnetization is allowed        to precess under the influence of the permanent gradient G₀ for        τ₁ units of time. At the end of the free precession period, a        180° refocusing pulse is applied, this produces a transverse        magnetization of the form:        m(x,y,z;1)=sin αρ(x,y,z)w(γg _(z)(x+z))e ^(−iγg) ^(z) ^((τ) ⁰        ^((x+z)+τ) ¹ ^(z))  (27)    -   5. The magnetization is now rewound in preparation for read-out:        the gradient field g_(z)G₁ is again turned on and the excited        magnetization is allowed to freely precess in the field        g_(z)G₁+G₀ for

$\tau_{0} + {\frac{1}{2}\tau_{1}}$units of time. To phase encode, the gradient field is turned on:

$G_{ph} = {\frac{{- 2}{\lambda\tau}_{1}}{{2\tau_{0}} + \tau_{1}}g_{z}{G_{2}.}}$At the conclusion of this free precession period, the transversemagnetization takes the form:

$\begin{matrix}{{m\left( {x,y,{z;2}} \right)} = {\sin\;{{\alpha\rho}\left( {x,y,z} \right)}{w\left( {\gamma\;{g_{z}\left( {x + z} \right)}} \right)}{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{1}{2}\gamma\; g_{z}{\tau_{1}{({z - x + {\lambda\; y}})}}}}} & (28)\end{matrix}$

-   -   6. To read out the magnetization, one immediately turns on the        field −g_(z)G₁. If t=0 at the start of the acquisition, then the        signal available for sampling is:

$\begin{matrix}{{S(t)} = {\int_{D}{\sin\;{{\alpha\rho}\left( {x,y,z} \right)}{w\left( {\gamma\;{g_{z}\left( {x + z} \right)}} \right)}{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{1}{2}\gamma\;{g_{t}{({\tau_{1} - t})}}{({z - x})}}{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{1}{2}\gamma\;\lambda\; g_{z}\tau_{1}y}{\mathbb{d}x}{\mathbb{d}y}{\mathbb{d}z}}}} & (29)\end{matrix}$One may change variables in this integral, letting a=z+x and b=z−x, toobtain:

$\begin{matrix}{{S(t)} = {\frac{1}{2}{\int_{D}{\sin\;\alpha\;{\rho\left( {\frac{a - b}{2},y,\frac{a + b}{2}} \right)}{w\left( {\gamma\; g_{z}a} \right)}{\mathbb{e}}^{{- {\mathbb{i}}}\frac{1}{2}\gamma\;{g_{z}{({\tau_{1} - t})}}b}{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{1}{2}\gamma\;\lambda\; g_{z}\tau_{1}y}{\mathbb{d}a}{\mathbb{d}y}{\mathbb{d}b}}}}} & (30)\end{matrix}$This can be interpreted as the 2-dimensional Fourier transform of theslice average:

$\begin{matrix}{{\overset{\_}{\rho}\left( {y,b} \right)} = {\frac{1}{2}{\int_{\frac{f_{0} - {\Delta\; f}}{\gamma\; g_{z}}}^{\frac{f_{0} + {\Delta\; f}}{\gamma\; g_{z}}}{{\rho\left( {\frac{a - b}{2},y,\frac{a + b}{2}} \right)}{w\left( {g_{z}a} \right)}{{\mathbb{d}a}.}}}}} & (31)\end{matrix}$

The slice averaging is illustrated in FIG. 7. The signal is therefore:

$\begin{matrix}{{S(t)} = {\int{{\overset{\_}{\rho}\left( {y,b} \right)}{\mathbb{e}}^{{- {\mathbb{i}}}\frac{1}{2}\gamma\;{g_{z}{\lbrack{{{({\tau_{1} - t})}b} + {\lambda\;\tau_{1}y}}\rbrack}}}{\mathbb{d}y}{{\mathbb{d}b}.}}}} & (32)\end{matrix}$At time t the signal is ρ(k(t)), where:

${k(t)} = {\frac{\gamma\; g_{z}}{2}{\left( {{\lambda\;\tau_{1}},\left( {\tau_{1} - t} \right)} \right).}}$This pulse sequence is illustrated in the timing diagram shown in FIG. 8and an image of a phantom using this approach is shown in FIG. 9.

There are several possible variations in this approach that lead to thesame result. For example, one could refocus immediately following theselective excitation. After the refocusing pulse, the magnetizationwould be allowed to freely precess, under the influence of the permanentgradient alone for 2τ₀ time units. At the conclusion of this freeprecession period, the magnetization could again be refocused. Thesignal would then be read out with read-out gradient equal to −g_(z)G₁,as before. At the start of the read out, the transverse magnetizationwould equal:

$\begin{matrix}{{m\left( {x,y,{z;2^{\prime}}} \right)} = {\sin\;\alpha\;{\rho\left( {x,y,z} \right)}{w\left( {\gamma\;{g_{z}\left( {x + z} \right)}} \right)}{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{1}{2}\gamma\; g_{z}{\tau_{0}{({z - x + {\lambda\; y}})}}}}} & (33)\end{matrix}$Weak Transverse Gradients (G₀>>G₁)

What makes the previous case especially simple is the assumption thatthe apparatus can produce adjustable gradients, with the strength of thefield gradient at least equal to the strength of the permanent fieldgradient. This is by no means necessary for the method to succeed. Theonly modification is in the definition of the slice average ρ(y,b). Thismore general case is now described.

The following is an illustrative case, with many possible variationsthat will not be spelled out. Suppose that one can generate adjustablefield gradients that are smaller than the magnitude of the permanentfield gradient in B₀. As before it is assumed that:G₀=(*,*,g_(z)z),G₁=(*,*,x) and G₂=(*,*,y),and that one can generate ρ₁G₁+ρ₂G₂ where |η₁|,|η₂|<m_(g)<<g_(z). Thesame steps are followed as described above. For the slice selectgradient, one can use G₀+g_(x)G₁, as before one can use λg_(x)G₂ forphase encoding and G₀−g_(x)G₁ as the read out gradient. Supposing that ascheme is used with two refocusing pulses, the calculations above arenot repeated but one skilled in the art will appreciate that the signalequation now reads:

$\begin{matrix}{{S(t)} = {\int_{D}{\sin\;\alpha\;{\rho\left( {x,y,z} \right)}{w\left( {\gamma\;{g_{z}\left( {x + z} \right)}} \right)}{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{1}{2}\gamma\;{g_{z}{({\tau_{0} - t})}}{({{g_{z}z} - {g_{x}x}})}}{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{1}{2}\gamma\;\lambda\; g_{x}\tau_{0}y}{\mathbb{d}x}{\mathbb{d}y}{\mathbb{d}z}}}} & (34)\end{matrix}$This can also be interpreted as a 2d-Fourier transform of an averageover the excited slice. The only difference between this case and theprevious case is that the average is over lines that meet the slice at afixed angle, not necessarily 90°. This is shown in FIG. 10.

To see this analytically, the variables are changed setting:

$\begin{matrix}{a = {{\frac{{g_{z}z} - {g_{x}x}}{g}\mspace{14mu} b} = \frac{{g_{z}x} + {g_{x}z}}{g}}} & (35)\end{matrix}$where g=√{square root over (g_(x) ²+g_(z) ²)}. This gives:

$\begin{matrix}\begin{matrix}{{S(t)} = {\int_{D}{\sin\;{{\alpha\rho}\left( {\frac{{g_{z}b} - {g_{x}a}}{g},y,\frac{{g_{z}b} - {g_{x}b}}{g}} \right)}}}} \\{w\left( {\frac{\gamma}{g}\left( {{\left( {g_{z}^{2} - g_{x}^{2}} \right)a} + {2g_{z}g_{x}b}} \right)} \right)} \\{{\mathbb{e}}^{{- {\mathbb{i}}}\frac{1}{2}\gamma\;{g{({\tau_{0} - t})}}a}{\mathbb{e}}^{{- {\mathbb{i}}}\frac{1}{2}\gamma\;\lambda\; g_{x}\tau_{0}y}\ {\mathbb{d}b}{\mathbb{d}y}{\mathbb{d}a}}\end{matrix} & (36)\end{matrix}$Changing variables in the b-integral, setting:

$\begin{matrix}{s = {{\frac{{\left( {g_{z}^{2} - g_{x}^{2}} \right)a} + {2g_{x}g_{z}b}}{g^{2}}\mspace{25mu} b} = \frac{{\left( {g_{x}^{2} - g_{z}^{2}} \right)a} + {g^{2}s}}{2g_{x}g_{z}}}} & (37)\end{matrix}$one obtains:S(t)=∫ ρ(a,y)e ^(−iγ(g(τ) ⁰ ^(−t)a+λτ) ⁰ ^(gy]) dyda.  (38)where:

$\begin{matrix}{{\overset{\_}{\rho}\left( {a,y} \right)} = {\frac{g^{2}}{2g_{x}g_{z}}{\int{{\rho\left( {\frac{\left( {s - a} \right)}{2g_{x}},y,\frac{g\left( {s + a} \right)}{2g_{z}}} \right)}{w\left( {g\;\gamma\; s} \right)}{\mathbb{d}s}}}}} & (39)\end{matrix}$Thus, ρ is ρ averaged along lines with slope (g_(z),0,g_(x)). These arelines orthogonal to the read-out direction (−g_(x),0,g_(z)), which arenot, in general, parallel to the slice select direction (g_(x),0,g_(z)).In the case at hand, the lines over which ρ is averaged make an angle θwith the slice select direction where:

$\begin{matrix}{{\cos\;\theta} = {\frac{2g_{x}g_{z}}{g_{x}^{2} + g_{z}^{2}}.}} & (40)\end{matrix}$The length of the intersection of these lines with the selected slice isminimized when θ=90°, hence this length is slowly varying for θ close to90°.The Dependence of SNR on the Permanent Gradient Strength

The signal-to-noise ratio attainable using the procedures describedabove is now considered. The analysis of the SNR is essentially the sameas it would be in a standard 2-dimensional imaging system. The maineffects of a large permanent gradient are to reduce the physicalthickness of the excited slice, thereby reducing the signal, as well asto reduce the allowable spacing, Δt, between the acquisition ofsuccessive samples, thereby increasing the receiver bandwidth. Supposethat one has a slice thickness d and a rectangular field-of-view of sizeL. If Δx denotes the (isotropic) pixel length, Δt is the acquisitiontime and N-samples are collected in each direction, then:SNR∝dΔx²N√{square root over (Δt)}.  (41)

It is desired to understand how the strength of the permanent gradientand the ratio ν=g_(x)/g_(z) affects the SNR. This brings the FOV intothe equation, along with the gradient strength. Suppose that: g=√{squareroot over (|G₀|²+|G₁|²)}, then, to avoid aliasing, one would need totake:

$\begin{matrix}{{\Delta\; t} \leq {\frac{\sqrt{1 + v^{2}}}{\gamma\; g_{x}L}.}} & (42)\end{matrix}$Using that L=NΔx, combining (41) and (42) one obtains:

$\begin{matrix}{{SNR} \propto {d\;\Delta\; x{\sqrt{\frac{L\left( {1 + v^{2}} \right)}{g_{x}}}.}}} & (43)\end{matrix}$Thus, the SNR is not directly affected by the presence of a strongpermanent background gradient. It is indirectly affected, because inorder to get the desired resolution the maximum frequency sampled ink-space must satisfy:

$\begin{matrix}{k_{\max} = {\frac{1 + v^{2}}{2{d\left( {1 - v^{2}} \right)}}.}} & \;\end{matrix}$

If ν is close to zero (g_(x)=g_(z)), then a thin slice is needed to gethigh resolution. This has the effect of lowering the SNR and wouldnecessitate using several averages of each line. Beyond this, a largegradient also may lead to a thin slice, if one is constrained in theamount of RF-power one may apply. This also diminishes the SNR. Theseeffects are illustrated in FIGS. 12 and 13, which show images of apomegranate obtained using different values for ν. FIG. 12 shows imagesof a pomegranate made using the slant-slice protocol with various valuesof the ratio

$v = {\frac{g_{x}}{g_{z}}.}$In FIG. 12, the geometric distortion caused by the slant of the slicehas not been corrected, whereas in FIG. 13 it has been. The applied(readout and slice select) gradients have amplitude 3 mT/m, and theamplitude of the permanent gradient ranged between 3 and 18 mT/m. Theslice thickness in Hertz is kept fixed throughout these images, which inturn means that the slice thickness in mm decreases as the gradientstrength increases. All images have the same intrinsic resolution. TheSNR decrease is caused by the decrease in signal due to the thinning ofthe slice. The following parameters were used for the scan in FIG. 12:TE=12 ms, TR=400 ms, 256 PE steps, Scan time=102 sec., FOV=256×256 mm².

FIG. 13 shows images from FIG. 12 with the corrections for the geometricdistortion.

On the other hand, as noted by Rose and Crowley, a strong permanentgradient causes k-space to be rapidly traversed and so one can, inprinciple, refocus the transverse magnetization and reread the same lineseveral times. Indeed, the time to traverse a line in k-space isproportional to the strength of the read-out gradient. Hence, if ν isnot too small, then within a single repeat time, one could refocus themagnetization and reread the line a number of times proportional to g,and thereby regain some of the lost SNR, without any increase in imagingtime. In FIG. 14 we show images where the ratio between G₀ and G₁ equals1, but their strengths are simultaneously increased. The slice thickness(in mm) is held constant. Notice the moderate decrease in the SNR as thegradient strength is increased. All images have the same intrinsicresolution.

The Resolution in the Linear Case

If one can generate an adjustable gradient of strength equal to that ofthe permanent gradient, then the pixels are rectangular and theresolution is determined by the usual heuristic formula:

$\begin{matrix}{{\Delta\; x} \approx {\frac{1}{k_{\max}}.}} & (44)\end{matrix}$If the maximum adjustable gradient |G₁| is smaller than the permanentgradient, |G₀|, then there is additional averaging involved in signalacquisition. To quantify this effect one can make the followingsimplifying assumption: the spin density ρ is constant along linesparallel to the slice select direction. Indeed, this is also a “worstcase” analysis, when comparing the slant slice protocol to a protocolwith averaging parallel to the slice direction (i.e. |G₀|=|G₁|). Thisassumption is reasonable for thin slices and a slowly varying spindensity. In this case, at least within the excited slice, one gets:

$\begin{matrix}{{\rho\left( {x,z} \right)} \approx {{f\left( \frac{{- {xg}_{z}} + {zg}_{x}}{g} \right)}.}} & (45)\end{matrix}$One uses g_(x)|G₁|,g_(z)=|G₀|, to simplify the notation. To simplify theanalysis, one may ignore the third dimension, which would, in any casebe obtained by phase encoding in a direction orthogonal to the planespanned by G₀ and G₁.

Ignoring the third dimension, the signal equation becomes:

$\begin{matrix}{{{S(t)} = {\int_{- \frac{L}{2}}^{\frac{L}{2}}{\left( \frac{g^{2}}{2g_{x}g_{z}} \right){\int{{f\left( {{a\frac{g^{2}}{2g_{x}g_{z}}} - {s\frac{g_{z}^{2} - g_{x}^{2}}{2g_{x}g_{z}}}} \right)}{w\left( {g\;\gamma\; s} \right)}{\mathbb{d}s}\;{\mathbb{e}}^{{- 2}\pi\;{ika}}\ {\mathbb{d}a}}}}}},} & (46)\end{matrix}$where 2πk=γ(τ₀−t)g. Letting:

$\sigma = {s\frac{g_{z}^{2} - g_{x}^{2}}{2g_{x}g_{z}}}$obtains:

$\begin{matrix}{{{S(t)} = {\int_{- \frac{L}{2}}^{\frac{L}{2}}{\left( \frac{g_{z}^{2} + g_{x}^{2}}{g_{z}^{2} - g_{x}^{2}} \right){\int{{f\left( {{a\frac{g^{2}}{2g_{x}g_{z}}} - \sigma} \right)}{w\left( {\gamma\;{g\left( \frac{2\sigma\; g_{x}g_{z}}{g_{z}^{2} - g_{x}^{2}} \right)}} \right)}{\mathbb{d}\sigma}\;{\mathbb{e}}^{{- 2}\pi\;{ika}}\ {\mathbb{d}a}}}}}},} & (47)\end{matrix}$One final change of variables, gives this integral a very simpleinterpretation:

$\begin{matrix}{{\alpha = {a\frac{g^{2}}{2g_{x}g_{z}}}},} & (48)\end{matrix}$obtains:

$\begin{matrix}{{{S(t)} = {\int_{- \frac{L}{2}}^{\frac{L}{2}}{\left( \frac{2g_{x}g_{z}}{g_{z}^{2} - g_{x}^{2}} \right){\int{{f\left( {\alpha - \sigma} \right)}{w\left( {\gamma\;{g\left( \frac{2\sigma\; g_{x}g_{z}}{g_{z}^{2} - g_{x}^{2}} \right)}} \right)}{\mathbb{d}\sigma}\;{\mathbb{e}}^{{- 2}\pi\; i\frac{g_{x}g_{z}}{g^{2}}}\ {\mathbb{d}\alpha}}}}}},} & (49)\end{matrix}$

The measurement is the Fourier transform, at frequency

${\frac{2g_{x}g_{z}}{g^{2}}k},$of the convolution of ƒ with:

$\begin{matrix}{{W(\sigma)} = {\left( \frac{2g_{x}g_{z}}{g_{z}^{2} - g_{x}^{2}} \right){{w\left( {\gamma\; g\frac{2\sigma\; g_{x}g_{z}}{g_{z}^{2} - g_{x}^{2}}} \right)}.}}} & \left( {50a} \right)\end{matrix}$Thus the slanted slice has three different effects on the resolution:

-   -   1. It reduces the effective maximum frequency sampled by a        factor of

$\frac{2g_{x}g_{z}}{g^{2}}$and scales the sample spacing in k-space by the same factor:

$\begin{matrix}{k_{\max,{ro}} = {{\frac{2g_{x}g_{z}}{g^{2}}k_{\max}\mspace{14mu}{and}\mspace{14mu}\Delta\; k_{ro}} = {\frac{2g_{x}g_{z}}{g^{2}}\Delta\; k}}} & \left( {50b} \right)\end{matrix}$

-   -   2. It causes blurring due to the convolution with W along the        slanted line. As g_(x) approaches g_(z) the convolution        approaches convolution with a scaled delta function.    -   3. If the angle θ is close to zero, so that the effect of the        convolution with W cannot be removed, then the effective field        of view is the support of W*ƒ, rather than the support of ƒ.        The effect of the convolution can, in principle, be removed if        the Fourier transform of W does not vanish in the interval

$\left\lbrack {{{- \frac{2v}{1 + v^{2}}}k_{\max}},{\frac{2v}{1 + v^{2}}k_{\max}}} \right\rbrack.$Suppose that w(s)=χ_([−γd,γd])(s), then

$\begin{matrix}{{\hat{W}(k)} = {\frac{2v}{{\pi\left( {1 - v^{2}} \right)}k}{{\sin\left( {\pi\;{dk}\frac{1 - v^{2}}{2v}} \right)}.}}} & \left( {50c} \right)\end{matrix}$Hence, the effect of the slant slice convolution can be removed, withoutexcessive amplification of the noise, for frequencies that satisfy:

$\begin{matrix}{k{{\operatorname{<<}\frac{v}{d\left( {1 - v^{2}} \right)}}.}} & (51)\end{matrix}$Recalling that k_(max) is also scaled, the resolution is effectivelygiven by

$\begin{matrix}{{{\Delta\; x} \approx {\frac{1 + v^{2}}{2v}\frac{1}{k_{\max}}}},} & (52)\end{matrix}$provided:

$\begin{matrix}{k_{\max} < {\frac{1}{2d}{\frac{1 + v^{2}}{1 - v^{2}}.}}} & (53)\end{matrix}$Note that k_(max)=γNΔtg if 2N+1 samples are collected. This shows thatthe resolution in the readout direction is effectively determined byg_(x):

${\Delta\; x} \approx {\frac{\sqrt{1 + v^{2}}}{2\;\gamma\; N\;\Delta\;{tg}_{x}}.}$Putting together the two formula shows that this approach has aneffective resolution limit:

$\begin{matrix}{{{{\Delta\; x_{\lim}} \approx {d\frac{1 - v^{2}}{v}}} = {2d\frac{\sin\;\theta}{\cos\;\theta}}},} & (54)\end{matrix}$where θ is the angle between the slice select direction and thedirection along which the spin density is averaged. With this modality,it may be desirable to use thin slices, measured many times. Inprinciple, this would allow the recovery of any lost resolution, thoughat the cost of additional acquisition time.Inhomogeneous Fields ithout Critical Points, General Case

The general case of imaging with an inhomogeneous field without criticalpoints is now considered. As before, the permanent gradient in thebackground field may be used as a slice select gradient, or part of aslice select gradient. Several investigators have obtained partialresults in this direction. In the prior art, the analysis is eitherperturbative, or done with unnecessarily restrictive hypotheses on thebackground field or the gradients. In this embodiment, the inventorsprovide minimal hypotheses on the background and gradient fields underwhich the measurements can, after a change of variables in physicalspace, be interpreted as samples of the ordinary Fourier transform.

The following notational conventions will be used: suppose that theobject being imaged lies in a region of space that is denoted by D,where D is the field-of-view. The object is described by a densityfunction ρ(x,y,z), supported in D. As usual B₀ denotes the backgroundfield.

-   -   1. For purposes of the present description:        φ₀(x,y,z)=|B₀(x,y,z)|,        and suppose it takes values in [c₀,c₁], for points lying in D.        The local Larmor frequency at (x,y,z) is γφ₀(x,y,z).    -   2. If ƒ is a real valued function defined in a region D, then        for each c∈R:        ƒ⁻¹(c)={(x,y,z)∈D:ƒ(x,y,z)=c}        The initial assumptions concern the function φ₀(x,y,z):    -   (a) The function φ₀(x,y,z) has no critical points within the        field-of-view. This means that the level sets:        S _(γc)={(x,y,z)∈D:φ ₀(x,y,z)=c}  (55)        are smooth. The level sets are labeled by the local Larmor        frequency.    -   (b) It is assumed that the coordinates (x,y,z) are chosen so        that each level set S_(γc) can be represented as a graph over a        (fixed) region R in the (x,y)-plane. In other words, there is a        smooth function z(x,y,c) so that φ₀(x,y,z(x,y,c))=c for        c∈[c₀,c₁] and therefore:        S _(γc)={(x,y,z(x,y,c)):(x,y)∈R}  (56)        With these assumptions, the region D is the set:        D={(x,y,z(x,y,c)):(x,y)∈R and ∈[c ₀ ,c ₁]}.        The second assumption is not strictly necessary. Though, without        something like it, one cannot expect to use the Fourier        transform, in any simple way, to reconstruct the spin density.        Indeed the mathematical analysis required becomes vastly more        complicated.

A formula will now be provided for the measured signal withoutadditional gradients, which will later be modified to include the effectof gradients. Let ω₀ denote γφ₀(0,0,0), and assume that [ω₀−Δω,ω₀+Δω] iscontained in [γc₀,γc₁]. Suppose that the polarized sample is irradiatedwith a selective RF-pulse designed to flip spins lying in the regionwhere the local Larmor frequency lies between ω₀−Δω and ω₀+Δω. This canbe described in terms of an excitation profile w(φ₀). For example, anideal 90°-flip has excitation profile given by:w ₉₀(c)=1 for γc∈[ω ₀−Δω,ω₀+Δω]  (57)w ₉₀(c)=0 for γc∈[ω ₀−Δω,ω₀+Δω].  (58)After the initial RF-pulse, the signal as a function of time is givenby:

$\begin{matrix}\begin{matrix}{{S_{0}(t)} = {C\;\gamma{\int_{D}{{\rho\left( {x,y,z} \right)}{b_{1{rec}}\left( {x,y,z} \right)}\phi_{0}^{2}}}}} \\{\left( {x,y,z} \right){w\left( {\phi_{0}\left( {x,y,z} \right)} \right)}\ {\mathbb{d}x}{\mathbb{d}y}{\mathbb{d}z}}\end{matrix} & (59)\end{matrix}$One factor of φ₀(x,y,z) comes from the definition of the equilibriummagnetization and the other comes from Faraday's Law. Time is labeled sothat t=0 corresponds to the spin echo induced by a refocusing pulse asdescribed in the previous section. Using the second assumption, thevariables are changed to (x,y,c), obtaining:

$\begin{matrix}{{S_{0}(t)} = {C\;\gamma{\int_{R}{\int_{c_{0}}^{c_{1}}{\frac{{\rho\left( {x,y,c} \right)}{b_{1{rec}}\left( {x,y,c} \right)}}{\partial_{z}{\phi_{0}\left( {x,y,{z\left( {x,y,c} \right)}} \right)}}\ c^{2}{w(c)}{\mathbb{d}c}\ {\mathbb{d}x}{\mathbb{d}y}}}}}} & (60)\end{matrix}$The denominator in equation (60) is computed using the chain rule. It isthe Jacobian of the transformation (x,y,z)→φ₀(x,y,z). To simplify thenotation ρ(x,y,c) and b_(rec1)(x,y,c) denote ρ(x,y,z(x,y,c)) andb_(1rec)(x,y,z(x,y,c)) respectively.

As above, demodulating and averaging S₀(t) over a sufficiently smalltime interval [−τ_(acq),τ_(acq)] gives:

$\begin{matrix}{\begin{matrix}{{\overset{\_}{S}(0)} = {C\;\gamma{\int_{R}{\int_{c_{0}}^{c_{1}}\frac{{\rho\left( {x,y,c} \right)}{b_{1{rec}}\left( {x,y,c} \right)}}{\partial_{z}{\phi_{0}\left( {x,y,{z\left( {x,y,c} \right)}} \right)}}}}}} \\{c^{2}{w(c)}\sin\;{c\left( {\tau_{acq}\left( {{\gamma\; c} - \omega_{0}} \right)} \right)}{\mathbb{d}c}{\mathbb{d}x}{\mathbb{d}y}}\end{matrix}\ } & (61)\end{matrix}$If Δω is small enough, then:

$\begin{matrix}{{\overset{\_}{S}(0)} \approx {C{\int_{R}{\frac{{\rho\left( {x,y,{\gamma^{- 1}\omega_{0}}} \right)}{b_{{rec}\; 1}\left( {x,y,{\gamma^{- 1}\omega_{0}}} \right)}}{\partial_{z}{\phi_{0}\left( {x,y,{z\left( {x,y,{\gamma^{- 1}\omega_{0}}} \right)}} \right)}}\ {\mathbb{d}x}{\mathbb{d}y}}}}} & (62)\end{matrix}$Up to a scale factor, this is the total spin density along the sliceS_(ω) ₀ , weighted by b_(rec1)/∂_(z)φ₀.

The addition of gradients to resolve the 3-dimensional structure of ρ isnow considered. As in the previous section, a scheme may be used thatdirectly samples a 3-dimensional Fourier transform. Because thedirection of B₀ varies, one needs to consider the properties of theactual gradient fields, and not simply their projections along a fixeddirection. For this discussion, it is supposed that one can generategradient fields in the region D, which can be represented as linearcombinations of two basic gradient fields denoted by G₁ and G₂. They aresolutions, defined in a neighborhood of D, of the time independent,vacuum Maxwell equation.

As before, in order to do 3D-imaging in a straightforward manner werequire two further assumptions about the gradient fields:

-   -   3. For all pairs (η₁,η₂) satisfying |η_(j)≦η_(max), j=1, 2, one        can generate the field η₁G₁+η₂G₂.    -   4. For each c∈[c₀,c₁], the map from R to a subset of the plane,        defined by:

$\begin{matrix}{X = \frac{\left\langle {{G_{1}\left( {x,y,z} \right)},{B_{0}\left( {x,y,z} \right)}} \right\rangle}{\phi_{0}\left( {x,y,z} \right)}} & (63) \\{Y = \frac{\left\langle {{G_{2}\left( {x,y,z} \right)},{B_{0}\left( {x,y,z} \right)}} \right\rangle}{\phi_{0}\left( {x,y,z} \right)}} & (64)\end{matrix}$is one-to-one and has a smooth inverse

In standard imaging, with a background field given by B₀=(0,0,b₀), thebasic gradient fields are modeled as G₁=(z,0,x) and G₂=(0,z,y). It iswell known that one can generate the linear combinations described inassumption 3 above. In this case X=x, Y=y, so both assumptions areeasily seen to be satisfied.

It will be appreciated by those skilled in the art that due to thelinear nature of Maxwell's equations, if, using electromagnets, one cangenerate the fields G₁ and G₂, in the region D, then, by adjusting thecurrents, one can also generate the linear combinations called for inassumption 1. In and of itself, condition 3 is a consequence ofMaxwell's equation. It is also the fundamental requirement for obtainingdata that can be interpreted as samples of the Fourier transform of afunction simply related to ρ.

Assumption 4 is a bit harder to check in practice. If the direction ofB₀ does not vary too much over the region D, then this condition is alsoeasily satisfied. If {{tilde over (x)},{tilde over (y)},{tilde over(z)}} is an orthonormal frame such that B₀(0,0,0) is parallel to {tildeover (z)}, then it is easy to show that there are vacuum solutions ofMaxwell's equations of the form:G ₁ =x{tilde over (z)}+a ₁ {tilde over (x)}+a ₂ {tilde over (y)} G ₂=y{tilde over (z)}+a ₂ {tilde over (x)}+b ₂ {tilde over (y)},  (65)where a₁,a₂,b₁,b₂ are linear functions, vanishing at (0,0,0). With thesesolutions:X=x+h .o .t., Y=y+h .o .t.,here h .o .t . are terms vanishing quadratically at (0,0,0). Hence, ifthe field-of-view is not too large, or, alternately, the direction of B₀does not vary too rapidly, then the pair (X,Y), defined by the fieldsgiven in equations (63) and (64), satisfies assumption 4.

How the expression for the signal is modified by the addition of thegradient fields may now be seen. It is assumed that φ₀ is sufficientlylarge throughout the field of view so that one can ignore components ofG₁ and G₂ orthogonal to B₀. Here a spatial encoding scheme is used likethat described above. For each allowable pair (η₁, η₂) there is anexpression for the signal:

$\begin{matrix}{{S_{({\eta_{1},\eta_{2}})}(t)} \approx {\frac{C\;\omega_{0}^{2}}{\gamma}{\int_{\circ 2}{\int_{c_{0}}^{c_{1}}{\frac{{\rho\left( {x,y,c} \right)}{b_{1{rec}}\left( {x,y,c} \right)}}{\partial_{z}{\phi_{0}\left( {x,y,{z\left( {x,y,c} \right)}} \right)}}{{gw}(c)}{\mathbb{e}}^{{\mathbb{i}\gamma}{\lbrack{{c{({t - \tau_{3}})}} + {t\;\eta_{1}{X{({x,y,c})}}} + {t\;\eta_{2}{Y{({x,y,c})}}}}\rbrack}}{\mathbb{d}c}{\mathbb{d}x}{{\mathbb{d}y}.}}}}}} & (66)\end{matrix}$The time parameter is normalized so that the refocusing pulse ends att=0, at which time, the gradient field η₁G₁+η₂G₂ is switched on.

Using assumption 2, one can solve for x(X,Y,c) and y(X,Y,c) throughoutthe region of integration. Let J(X,Y,c) denote the Jacobian of thischange of variables: dxdy=J(X,Y,c)dXdY. The expression for the signalbecomes:

$\begin{matrix}{{S_{({\eta_{1},\eta_{2}})}(t)} \approx {\frac{C\;\omega_{0}^{2}}{\gamma}{\int_{c_{0}}^{c_{1}}{\int_{\circ 2}{{\overset{\_}{\rho}\left( {X,Y,c} \right)}{w(c)}{\mathbb{e}}^{{\mathbb{i}\gamma}{\lbrack{{c{({t - \tau_{3}})}} + {t\;\eta_{1}X} + {t\;\eta_{2}Y}}\rbrack}}{\mathbb{d}X}{\mathbb{d}Y}{{\mathbb{d}c}.}}}}}} & (67)\end{matrix}$where:

$\begin{matrix}{{\overset{\_}{\rho}\left( {X,Y,c} \right)} = {\frac{\begin{matrix}{\rho\left( {{x\left( {X,Y,c} \right)},{y\left( {X,Y,c} \right)},c} \right)b_{\;{1\;{rec}}}} \\\left( {{x\left( {X,Y,c} \right)},{y\left( {X,Y,c} \right)},c} \right)\end{matrix}}{\partial_{z}{\phi_{0}\left( {{x\left( {X,Y,c} \right)},{y\left( {X,Y,c} \right)},c} \right)}}{J\left( {X,Y,c} \right)}}} & (68)\end{matrix}$Demodulating and sampling one can measure samples F(w ρ)(k_(j)), where,as before, the points {k_(j)} lie along straight lines within C. Thenormalization here is a little different from that used above. In theearlier case, with B₀ given by equation (7), is was not necessary tochange variables in the “z-direction.”

Up to a constant, the measured signal equals the ordinary Fouriertransform of w ρ(X,Y,c). Using a variety of reconstruction techniques,based on the Fourier transform, such as regridding and filteredback-projection, one can therefore reconstruct w ρ(X,Y,c). To determinethe original spin density requires a knowledge of b_(1rec) and thetransformations:(x,y,z)⇄(x,y,c)⇄(X,Y,c).  (69)These in turn can be computed with a knowledge of the background fieldB₀ and the basic gradient fields G₁ and G₂. The necessary coordinatetransformations are determined by the fields B₀, G₁ and G₂, which meansthat they can be computed once and stored. But for the need to do thisreparameterization before displaying the image, the computationalrequirements for imaging with an inhomogeneous field are comparable tothose found in X-ray CT. Using the phase encoding-frequency encodingapproach described above, with uniform sample spacing, one could use astandard FFT for the reconstruction step.

The conditions above on the background field and gradient fields areessentially that the functions:

$\begin{matrix}{{\phi_{0}\left( {x,y,z} \right)},\frac{\left\langle {{G_{1}\left( {x,y,z} \right)},{B_{0}\left( {x,y,z} \right)}} \right\rangle}{\phi_{0}\left( {x,y,z} \right)},\frac{\left\langle {{G_{\; 2}\left( {x,\; y,\; z} \right)},\;{B_{\; 0}\left( {x,\; y,\; z} \right)}} \right\rangle}{\;{\phi_{\; 0}\left( {x,\; y,\; z} \right)}}} & (70)\end{matrix}$define a one-to-one map from the field-of-view to a set of the form[c₀,c₁]×R, with R a subset of i². Implicitly, it is assumed that thedirection of B₀ does not vary too much within D. This assumption isneeded in order to apply the analysis of selective excitation with aninhomogeneous background field presented in the afore-mentioned Epsteinpaper. In this case, one also expects the receive coil sensitivity,b_(rec1), to be approximately constant (or a least bounded from below)within D. Under these conditions it may be seen that the size of themeasured signal obtainable with an inhomogeneous field should becomparable to the signal that can be obtained with a homogeneous field.Because there is always a large gradient, diffusion effects may alsodiminish the signal.Analysis in the Non-Linear Case for the 2D Imaging Protocol

This section briefly describes the needed modifications, for the2D-slice embodiment, if the gradient fields G₀,G₁,G₂ are not linear butsatisfy the conditions enumerated above. For simplicity, we firstdescribe how this approach would be applied to image a 2-dimensionalobject, so that the slices are 1-dimensional. The modifications neededto image 3-dimensional object with 2-dimensional slices is described atthe end of this section. A 2D protocol is described using two refocusingpulses. In this case, B₀=B₀₀+G₀, where B₀₀ is the uniform fieldB₀₀=(0,b₀). If G denotes an adjustable gradient field, then the localLarmor frequency is determined by:

$\begin{matrix}{{{B_{0} + G}} = {b_{0} + \left\langle {\frac{{2B_{00}} + G_{0} + G}{2b_{0}},{G_{0} + G}} \right\rangle + {O\left( \frac{1}{b_{0}} \right)}}} & (71)\end{matrix}$This equation shows that the validity of the assumption that, for thepurposes of analyzing the MR-signal, the gradient fields can be replacedby their projections onto B₀, is equivalent to the assumption that:|G ₀ +G|<<b ₀.  (72)This assumption pertains throughout the calculations that follow.

Modifying the notation in the linear case, then:

$\begin{matrix}{{g_{0} = \left\langle {\frac{B_{00}}{b_{0}},G_{0}} \right\rangle},\mspace{14mu}{g_{1} = {\left\langle {\frac{B_{00}}{b_{0}},G_{1}} \right\rangle.}}} & (73)\end{matrix}$The assumptions on the fields G₀, G₁ imply that ∇g₀, ∇g₁ are linearlyindependent at every point within the field of view.

As shown by Epstein in Magnetic Resonance Imaging in InhomogeneousFields, Inverse Problems, Vol. 20 (2004), pp. 753-780, provided thedirection of B₀ does not vary too much within the field-of-view, theselective excitation step proceeds very much as in the linear case.After the sample becomes polarized in the background field B₀, one mayturn on the field G₁ and expose the sample to a selective RF-pulse. Ifw(s) is the slice profile, then the magnetization at the conclusion ofthe α-RF-pulse is:m(0′)=sin αρ(x,z)w(γ(g ₀ +g ₁))e ^(iτ) ⁰ ^(γ(g) ⁰ ^(+g) ¹ ⁾.  (74)The transverse component is non-zero in the non-linear region of spacewhere:w(γ(g ₀(x,z)+g ₁(x,z)))≠0.  (75)After a refocusing pulse:m(1′)=sin αρ(x,z)w(γ(g ₀ +g ₁))e ^(−iτ) ⁰ ^(γ(g) ⁰ ^(+g) ¹ ⁾.   (76)The field G₁ is turned off and the magnetization is allowed to freelyprecess for 2τ₀ time units and is once refocused giving:m(2′)=sin αρ(x,z)w(γ(g ₀ +g ₁))e ^(−iτ) ⁰ ^(γ(g) ⁰ ^(−g) ¹ ⁾.   (77)

Finally, at t=0, the field −G₁ is again turned on to obtain the measuredsignal:

$\begin{matrix}{{S(t)} = {\int_{D}{\sin\;{{\alpha\rho}\left( {x,z} \right)}{w\left( {\gamma\left( {g_{0} + g_{1}} \right)} \right)}{\mathbb{e}}^{{{\mathbb{i}}{({t - \tau_{0}})}}{\gamma{({g_{0} - g_{1}})}}}{\mathbb{d}x}{{\mathbb{d}z}.}}}} & (78)\end{matrix}$Now using the basic assumptions, which imply that:gA=g ₀ +g ₁ gB=g ₀ −g ₁  (79)define coordinates throughout the region of space occupied by theobject, D, and define a map onto a region D′ of R² topologicallyequivalent to a square. Let dxdz=g²J(A, B)dAdB, the coefficient j isused to normalize so that g²J≈1 near the “center” of the slice. Thesignal equation becomes:

$\begin{matrix}\begin{matrix}{{S(t)} = {g^{2}{\int_{D}{\sin\;{{\alpha\rho}\left( {{x\left( {A,B} \right)},{z\left( {A,B} \right)}} \right)}}}}} \\{w\left( {\gamma\;{gA}} \right){\mathbb{e}}^{{{\mathbb{i}}{({t - \tau_{0}})}}\gamma\;{gB}}{J\left( {A,B} \right)}{\mathbb{d}A}{{\mathbb{d}B}.}}\end{matrix} & (80)\end{matrix}$For each fixed B,A a (x(A, B), z(A, B)) traces a smooth curve in thexz-plane which is, in some sense, transverse to the slice. This is, ofcourse, just the curve:g ⁻¹(g ₀(x,z)−g ₁(x,z))=B.Rewrite the signal as a 1-dimensional Fourier transform of the sliceaveraged function:ρ(B)=g ²∫ρ(x(A,B),z(A,B))w(γgA)J(A,B)dA  (81)provides:

$\begin{matrix}{{S(t)} = {\sin\;\alpha{\int_{D}{{\overset{\_}{\rho}(B)}{\mathbb{e}}^{{{\mathbb{i}}{({t - \tau_{0}})}}\gamma\;{gB}}{{\mathbb{d}B}.}}}}} & (82)\end{matrix}$

The measurements are then samples of the Fourier transform of ρ. Samplesof ρ can be reconstructed as a function of B. To reconstruct ρ in theslice defined −Δƒ≦g⁻¹γ(g₀+g₁)≦Δƒ, one only needs to invert the relationsin equation (79) to solve for (x,z) as functions of (A,B). Using thecomputation of J(A,B) one can also rescale the data according to thedensity of the individual slices. These steps are possible, at leastnumerically, if one knows the functions g₀(x,z),g₁(x,z).

For concreteness the example is considered where g₀=z²,g₁=x². Thefunctions A,B define coordinates in the half plane x>0:A=z²+x²,g₁=z²−x². The image of the positive quadrant is the region whereA>|B|. The area forms are related by the equation:

$\begin{matrix}{{dxdz} = {\frac{dAdB}{2\sqrt{A^{2} - B^{2}}}.}} & (83)\end{matrix}$FIG. 11 shows level lines of A and B in this quadrant. As illustrated,near to x=z the pixels are nearly rectilinear, but are less so near theaxes. A typical pixel is shaded. The function ρ is given by:

$\begin{matrix}{{\rho(B)} = {\int_{{- \Delta}\; f}^{\Delta\; f}{{w\left( {\gamma\; A} \right)}{\rho\left( {\sqrt{\frac{A - B}{2}},\sqrt{\frac{A + B}{2}}} \right)}{\frac{dA}{2\sqrt{A^{2} - B^{2}}}.}}}} & (84)\end{matrix}$From examination of FIG. 11, it is evident that the simple notions ofpixel and resolution, which are used with linear gradients, are notespecially meaningful in the strongly non-linear case. Indeed it isevident that resolution in the reconstructed image, when transformedback to physical coordinates, is unlikely to be either isotropic at mostpoints in the image plane, or homogeneous across the image.

Adding a third dimension is straightforward, given that one can generatetwo adjustable gradients G₁,G₂ so that the projections in theB₀-direction, g₀,g₁,g₂, define a smooth invertible mapping from thefield of view to a region in R³ topologically equivalent to a cube. Afield of the form G₀+g_(s)G₁ can be used for slice selection, multiplesof G₂ can be used to phase encode, and G₀−g_(s)G₁ can be used as aread-out gradient. As explained in the afore-mentioned Epstein article,the measurements obtained in this way can be interpreted, after a changeof physical (x-space) coordinates, as the Fourier transform ofnon-linear averages, of non-linear 2-dimensional slices of ρ(x,y,z).

An advantage of the 2D approach of the invention is that it allows theusage of a magnet with a substantial permanent gradient to be used asthe main magnet in an MR-imaging device. This is accomplished withoutsignificantly sacrificing either resolution, acquisition time or SNR. Byusing slanted slices one recovers, in almost its entirety, the formalismused to describe imaging with a homogeneous background field. Inparticular, one can use a simple FFT to reconstruct the image, alongwith a post-processing step to remove geometric distortions due toeither non-linear gradient fields or to the slant slice acquisition. Themethod of the invention produces high quality images, with acquisitiontimes comparable to what would be used in a standard imaging device.

The inventors have quantified the inherent limitations of the method asregards SNR and resolution, and neither seems, in any way insuperable.As noted above, given a constant physical slice thickness, a largepermanent gradient increases the noise in each acquired line in exactlyinverse proportion to the time required to acquire the line. Hence, byrefocusing and remeasuring these lines, one can recover all the lostSNR, without any increase in overall repeat time. If g_(x)<<g_(z), then,for a given measurement time, this approach does have an intrinsicallylower resolution than a standard imaging method, with a homogeneousbackground field. The only real constraint on the applicability of thismethod is that ν not be too small, which allows for an enormous increasein the latitude available to designers of practical, high resolution,time and SAR efficient MR-imaging systems.

This technique could be used to build a “one-sided” 3d MR-imaging systemas shown generally in FIG. 15, wherein the sample (patient) would lie ona patient table 100 to one side of the magnet 110. This could be usedfor open MR-systems, or specialized MR-systems, for example dental MR.In many of these applications, gradient and RF-coils 120 are situatednear to or around the sample, which is itself placed to one side of thestatic field generating magnet 110. As illustrated in FIG. 15, thegradient coils are controlled by gradient amplifier and controller 130,while the RF coils are connected to RF transmitter/receiver 140, whichprovides output to computer 150 for processing of the image data fordisplay on display device 160. In this configuration, one would not needto work against nature to design magnets with a very homogeneous field“outside the bore,” but can solve the easier problems associated withdesigning magnets that have fields with a moderate but smooth permanentgradient, which the approach of the invention uses to good advantage.One can even imagine how this technique could be applied to advantage inan application like well-logging. While direct inversion of themeasurements leads to images with geometric distortion, this distortionis completely determined by the field gradients. For a given magnet andgradient set, the geometric transformations, needed to remove thedistortion, could be computed once and stored.

Fields with Local Minima

If one could create a magnetic field, B₀ such that |B₀(x,y,z)| assumesan isolated nonzero minimum value, then one could measure localizedspectroscopic data using a single RF-pulse. Suppose the minimum occursat (x₀,y₀,z₀) and ω₀=γφ₀(x₀,y₀,z₀). A selective excitation which excitesfrequencies in the band [ω₀−Δω,ω₀+Δω] would only excite spins in theregion of space bounded by the closed surface:S _(ω) ₀ _(+Δω)={(x,y,z):γφ₀(x,y,z)=ω₀+Δω}.The entire FID is then produced by spins lying in a bounded region ofspace, close to the critical point of φ₀. Examples of such fields willnow be provided. These fields are obtained as perturbations of a uniformbackground field:B₀=[0,0,b₀].

-   -   For real parameters ε and δ:        B _(0,εδ) =B ₀ +ε[x,−y,0]+δ[−2xz,0,(z ² −x ²)]  (85)        It is an elementary computation to see that:        ∇×B _(0,εδ)=0,∇×B _(0εδ)=0        and therefore these vector fields define vacuum solutions of        Maxwell's equations. The length of B_(0,εδ) may be computed as:        |B _(0,εδ)|² =b ₀ ²+(ε²−2b ₀δ)x ²+ε² y ²+2b ₀ δz ²−4εδx ² z+δ        ²(x ² +z ²)²  (86)        The function φ₀ equals the square root of |B_(0,εδ)|², and        therefore:

${\nabla\phi_{0}} = {\frac{\nabla{B_{0,{ɛ\delta}}}^{2}}{2{B_{0,{ɛ\delta}}}}.}$The critical points of φ₀, where φ₀ does not vanish, therefore agreewith the critical points of |B_(0,εδ)|². At critical points, where φ₀≠0,the Hessian matrices satisfy:

$\begin{matrix}{H_{\phi_{0}} = \frac{H_{{B_{0,{ɛ\delta}}}^{2}}}{2{B_{0,{ɛ\delta}}}}} & (87)\end{matrix}$This shows that the types of the critical points agree as well.

From equation (86), it is clear that φ₀ has critical point at (0,0,0).The Hessian is a diagonal matrix:

$\begin{matrix}{{H_{\phi_{0}}\left( {0,0,0} \right)} = {\frac{2}{\phi_{0}\left( {0,0,0} \right)}\begin{pmatrix}{ɛ^{2} - {2b_{0}\delta}} & 0 & 0 \\0 & ɛ^{2} & 0 \\0 & 0 & {2b_{0}\delta}\end{pmatrix}}} & (88)\end{matrix}$This demonstrates the following result:

If δb₀>0 and ε²>2δb₀, then B_(0,εδ) is a vacuum solution of Maxwell'sequations such that B_(0,εδ) has an isolated minimum at (0,0,0).

The fields [x,−y,0] and [−2xz,0,z²−x²] are essentially standard gradientfields. Therefore B_(0,εδ) could be generated by using an arrangement ofgradient coils within a standard homogeneous, high field magnet. FIG. 6illustrates 2 level surfaces of B_(0,εδ) with b₀=1, ε=0.1, δ=0.0025.

SUMMARY

It has been shown how a magnet producing an inhomogeneous field, withoutcritical points, can be used to produce the background field for anMR-imaging system. In particular, the present invention illustrates thatthe main difficulty that one encounters with a inhomogeneous backgroundfield is that of refocusing the phase that accumulates along thedirection of ∇|B₀|. Several methods for directly sampling the 3d-Fouriertransform of ρ, or the 2d-Fourier transform of slices of ρ are outlined.

Simple geometric criteria are provided for the MR measurements made,using an inhomogeneous background field and nonlinear gradients, to besamples of the Fourier transform, up to a single change of coordinates.This coordinate change is determined by the background field, B₀ and thebasic gradient fields, G₁,G₂. As such it need only be computed once andstored. The computations strongly suggest that it should be possible toobtain a strong signal with a background field having substantialinhomogeneity.

By analyzing well-known imaging methods which employ fields with sweetspots, it may be shown that the principle underlying this approach isnothing other than the classical principle of stationary phase. In theprior art, these sweets spots are of saddle type. The aforementionedpaper of Epstein provides a simple geometric explanation for thedifficulty of obtaining localized information with critical points ofthis type: When the signal is large, the excited volume of space is seento be highly non-localized. Hence, it is only in the long time limit,when the signal has largely decayed, that spatially localizedinformation is available. Examples of fields are constructed such that|B₀| has an isolated, nonzero local minimum. Using such a field one canobtain a well localized excitation, ab initio, thereby avoiding the mostserious pitfall of earlier approaches to imaging with a sweet spot.

In the paper of Epstein cited above, the inventor shows that an isolatednon-zero, local minimum is not possible under various symmetryhypotheses. For translationally invariant fields, it may be seen thatlocal minima can occur, which, due to the translational invariance, mustoccur along a line. It is shown in the Epstein paper that an axiallysymmetric field cannot have an isolated minimum. Again, because of theaxial symmetry, such an isolated minimum would have to occur along theaxis of symmetry.

While the present invention has been described in connection with thepreferred embodiments of the various figures, it is to be understoodthat other similar embodiments may be used or modifications andadditions may be made to the described embodiment for performing thesame function of the present invention without deviating therefrom.Those skilled in the art will appreciate that there may be limits onmagnetic field inhomogeneity and the geometry of level sets near acritical point. There also may be limitations on RF-pulses ininhomogeneous fields as well as for critical points in fields withsymmetry. Such limitations are described in the Epstein paperincorporated by reference above. Therefore, the present invention shouldnot be limited to any single embodiment, but rather should be construedin breadth and scope in accordance with the appended claims.

1. A method of magnetic resonance imaging an object in the presence of apermanent background gradient, G₀, in the polarizing field, B₀,comprising: selecting a 2-dimensional slice of the object for excitationthat is not perpendicular to G₀; exciting the selected 2-dimensionalslice of the object by applying a selective RF-pulse in the presence ofa slice selection gradient G_(ss), where G_(ss) is a linear combinationof G₀ and a transverse gradient G_(ss) ^(app), generated as a linearcombination of basic gradient fields G₁, G₂ provided by a magneticresonance scanner; applying a readout gradient G_(re) where G_(re) is alinear combination of G₀ and a transverse gradient G_(re) ^(app)generated as a linear combination of G₁, G₂, where G_(ss) and G_(re) arenot parallel at any point in the selected excited slice; andreconstructing the selected excited slice of the object.
 2. A method asin claim 1, further comprising phase encoding the selected excited slicewith a gradient G_(ph) generated as a linear combination of G₁, G₂.
 3. Amethod as in claim 1, wherein the functions X=(G₁, B₀), Y=(G₂, B₀) andZ|B₀(x,y,z)|, define local coordinates that map the field of view of themagnetic resonance scanner onto a region of space.
 4. A method as inclaim 3, wherein G_(ss)=B₀+G₁ and G_(re)=B₀−G₁.
 5. A method as in claim1, wherein the step of applying the readout gradient G_(re) furthercomprises applying at least one refocusing pulse.
 6. A magneticresonance imaging device that images an object in the presence of apermanent background gradient, G₀, in the polarizing field, B₀,comprising: a magnetic resonance scanner that provides basic gradientfields G₁, G₂; an RF generator and RF coils that excite a selected2-dimensional slice of the object for excitation that is notperpendicular to G₀ by applying a selective RF-pulse in the presence ofa slice selection gradient G_(ss), where G_(ss) is a linear combinationof G₀ and a transverse gradient G_(ss) ^(app), generated as a linearcombination of the basic gradient fields G₁, G₂ provided by the magneticresonance scanner; a gradient generator and gradient coils that apply areadout gradient G_(re) where G_(re) is a linear combination of G₀ and atransverse gradient G_(re) ^(app) generated as a linear combination ofG₁, G₂, where G_(ss) and G_(re) are not parallel at any point in theselected excited slice; and a processor that reconstructs the selectedexcited slice of the object.
 7. A device as in claim 6, furthercomprising means for phase encoding the selected excited slice with agradient G_(ph) generated as a linear combination of G₁, G₂.
 8. A deviceas in claim 6, wherein the functions X=(G₁, B₀), Y=(G₂, B₀) andZ=∥B₀(x,y,z)∥, where G₀=ΔB₀, define local coordinates that map the fieldof view of the magnetic resonance scanner onto a region of space.
 9. Adevice as in claim 8, wherein G_(ss)=B₀+G₁ and G_(re)=B₀−G₁.
 10. Adevice as in claim 6, wherein the gradient generator applies the readoutgradient G_(re) by applying at least one refocusing pulse.